In expression (REF ), the \({\delta ^{\prime }}^2_{\eta i\left( r \right)}\)  parameter is related to the resonant frequency by the ratio (REF ), and the \({\delta ^{\prime }}^2_{\eta i}\)  parameter changes independently of the frequency of the spontaneous gamma-quantum. Note that the angular resonance width \({{\Upsilon }_{\eta i\left( r \right)}}\)  (REF ) increases significantly with increasing wave intensity. This is due to the growth of the function \(\mathbf {K} \left( {{r}_{\eta }} \right)\)  (REF ). So, for \(\eta \sim 1\)  a function \(\mathbf {K} \left( {{r}_{\eta }} \right)\sim 1\) , and for \(\eta \sim {{10}^{3}}\)  a function \(\mathbf {K} \left( {{r}_{\eta }} \right)\sim {{10}^{2}}\) . It should also be noted that in this paper we have limited the intensity of the laser field \(F\lesssim {{10}^{14}}\ {\text{V}}/{\text{cm}}\;\)  (see condition (REF ) and the text after this formula). Because of this, the angular resonance width \({{\Upsilon }_{\eta i\left( r \right)}}\)  will be significantly greater than the corresponding radiation corrections [1]}.
Given this, as well as Exps. (REF ), (REF ), (REF ), (REF ), (REF ), (REF ) after simple calculations [2]}, we obtain expressions for the resonant differential SB cross section for channels A and B in the following form:
\( \begin{split}\frac{d\sigma _{\eta i\left(l,r\right)}}{dx^{\prime }_{\eta i\left( r \right)}d{{\delta }^{\prime }}^{2}_{\eta i}}=\left( Z^2 \alpha r^2_e \right)\frac{J^2_{r-l}\left( \alpha _{\eta i\left(r \right)} \right)}{g^4_A} & \frac{\left(2\pi \right)^2 \left( 1-x^{\prime }_{\eta i\left( r \right)} \right)^3}{\left[ \left( 1+\eta ^2 \right)^2 \left({{\delta }^{\prime }}^{2}_{\eta i}-\delta ^{\prime 2}_{\eta i\left( r \right)} \right)+\Upsilon ^2_{\eta i\left( r \right)} \right]x^{\prime }_{\eta i\left( r \right)}} \times {} \\ & {} \times K\left( u_{\eta i\left( r \right)},\frac{r}{r_\eta } \right)d \delta ^{\prime 2}_{\eta f}d \varphi ^{\prime }_{-},\end{split} \) 
\( \begin{split}\frac{d\sigma _{\eta f\left( l,r \right)}}{dx^{\prime }_{\eta f\left( r \right)}d{{\delta }^{\prime }}^{2}_{\eta f}}=\left( Z^2\alpha r^2_e \right)K\left( u_{\eta f\left( r \right)},\frac{r}{r_{\eta }} \right) & \frac{\left(2\pi \right)^2\left(1-x^{\prime }_{\eta f\left( r \right)} \right)^{-1}}{\left[ \left( 1+\eta ^2 \right)^2 \left( {{\delta }^{\prime }}^{2}_{\eta f}-{{\delta }^{\prime }}^{2}_{\eta f\left( r \right)} \right)^2+\Upsilon ^2_{\eta f\left( r \right)} \right]x^{\prime }_{\eta f\left( r \right)}} \times {} \\ & {} \times \frac{J^2_{r-l}\left( \alpha _{\eta f\left( r \right)} \right)}{g^4_B}d{{\delta }^{\prime }}^{2}_{\eta i} d\varphi ^{\prime }_{-},\end{split} \) 
